第 18688 題：子空間直和

原題與已知條件

有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,13]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 有限維實向量空間、完整基底與exact rank證書。操作=directsum-multi-subspace-global；資料={"ambient_dimension":2,"subspace_bases":[[[1,0]],[[0,1]],[[1,13]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-direct-sum

逐步推導

source operation=directsum-multi-subspace-global;original snapshot and mathematics retained

explicit ordered source output fields=[pairwise_zero_intersections, total_basis_count, total_rank, global_direct_sum, covers_ambient, boundary]

directsum subspace0:complete ordered matrix=[[1], [0]];rows=2;columns=1

use exact invertible ROW operations only;original COLUMN roles/order stay fixed

directsum subspace0 pivot0 column0 normalized entry0 := div(1, 1)

directsum subspace0 pivot0 column0 normalized entry0 = 1

directsum subspace0:pivot row=0;column=0;nonzero pivot=1;old row=[1]

normalized full row=[1]

directsum subspace0:complete exact reduced matrix=[[1], [0]];pivot columns=[0]

rank=1;pivot columns independent and all original column relations preserved;zero rows retained

directsum subspace0:complete ORIGINAL independent columns=[[1, 0]]

declared subspace dimension=1;basis of its full real span;empty basis means zero subspace

directsum subspace1:complete ordered matrix=[[0], [1]];rows=2;columns=1

use exact invertible ROW operations only;original COLUMN roles/order stay fixed

directsum subspace1:matrix=[[0], [1]];selected row=1

swap rows0,1;matrix=[[1], [0]]

directsum subspace1 pivot0 column0 normalized entry0 := div(1, 1)

directsum subspace1 pivot0 column0 normalized entry0 = 1

directsum subspace1:pivot row=0;column=0;nonzero pivot=1;old row=[1]

normalized full row=[1]

directsum subspace1:complete exact reduced matrix=[[1], [0]];pivot columns=[0]

rank=1;pivot columns independent and all original column relations preserved;zero rows retained

directsum subspace1:complete ORIGINAL independent columns=[[0, 1]]

declared subspace dimension=1;basis of its full real span;empty basis means zero subspace

directsum subspace2:complete ordered matrix=[[1], [13]];rows=2;columns=1

use exact invertible ROW operations only;original COLUMN roles/order stay fixed

directsum subspace2 pivot0 column0 normalized entry0 := div(1, 1)

directsum subspace2 pivot0 column0 normalized entry0 = 1

directsum subspace2:pivot row=0;column=0;nonzero pivot=1;old row=[1]

normalized full row=[1]

directsum subspace2 eliminate row1 pivot0 entry0 product := mul(13, 1)

directsum subspace2 eliminate row1 pivot0 entry0 product = 13

directsum subspace2 eliminate row1 pivot0 entry0 difference := sub(13, 13)

directsum subspace2 eliminate row1 pivot0 entry0 difference = 0

directsum subspace2:target row=1;old=[13];factor=13;pivot row=[1]

new full row=[0]

directsum subspace2:complete exact reduced matrix=[[1], [0]];pivot columns=[0]

rank=1;pivot columns independent and all original column relations preserved;zero rows retained

directsum subspace2:complete ORIGINAL independent columns=[[1, 13]]

declared subspace dimension=1;basis of its full real span;empty basis means zero subspace

directsum pair0,1:complete ordered matrix=[[1, 0], [0, 1]];rows=2;columns=2

use exact invertible ROW operations only;original COLUMN roles/order stay fixed

directsum pair0,1 pivot0 column0 normalized entry0 := div(1, 1)

directsum pair0,1 pivot0 column0 normalized entry0 = 1

directsum pair0,1 pivot0 column0 normalized entry1 := div(0, 1)

directsum pair0,1 pivot0 column0 normalized entry1 = 0

directsum pair0,1:pivot row=0;column=0;nonzero pivot=1;old row=[1, 0]

normalized full row=[1, 0]

directsum pair0,1 pivot1 column1 normalized entry0 := div(0, 1)

directsum pair0,1 pivot1 column1 normalized entry0 = 0

directsum pair0,1 pivot1 column1 normalized entry1 := div(1, 1)

directsum pair0,1 pivot1 column1 normalized entry1 = 1

directsum pair0,1:pivot row=1;column=1;nonzero pivot=1;old row=[0, 1]

normalized full row=[0, 1]

directsum pair0,1:complete exact reduced matrix=[[1, 0], [0, 1]];pivot columns=[0, 1]

rank=2;pivot columns independent and all original column relations preserved;zero rows retained

directsum pair0,1 basis counts := sum(1, 1)

directsum pair0,1 basis counts = 2

directsum pair0,1 intersection dimension := sub(2, 2)

directsum pair0,1 intersection dimension = 0

pair original subspaces=[0, 1];individual independent basis counts=[1, 1];combined rank=2

dim intersection=0;pair intersection zero=是

directsum pair0,2:complete ordered matrix=[[1, 1], [0, 13]];rows=2;columns=2

use exact invertible ROW operations only;original COLUMN roles/order stay fixed

directsum pair0,2 pivot0 column0 normalized entry0 := div(1, 1)

directsum pair0,2 pivot0 column0 normalized entry0 = 1

directsum pair0,2 pivot0 column0 normalized entry1 := div(1, 1)

directsum pair0,2 pivot0 column0 normalized entry1 = 1

directsum pair0,2:pivot row=0;column=0;nonzero pivot=1;old row=[1, 1]

normalized full row=[1, 1]

directsum pair0,2 pivot1 column1 normalized entry0 := div(0, 13)

directsum pair0,2 pivot1 column1 normalized entry0 = 0

directsum pair0,2 pivot1 column1 normalized entry1 := div(13, 13)

directsum pair0,2 pivot1 column1 normalized entry1 = 1

directsum pair0,2:pivot row=1;column=1;nonzero pivot=13;old row=[0, 13]

normalized full row=[0, 1]

directsum pair0,2 eliminate row0 pivot1 entry0 product := mul(1, 0)

directsum pair0,2 eliminate row0 pivot1 entry0 product = 0

directsum pair0,2 eliminate row0 pivot1 entry0 difference := sub(1, 0)

directsum pair0,2 eliminate row0 pivot1 entry0 difference = 1

directsum pair0,2 eliminate row0 pivot1 entry1 product := mul(1, 1)

directsum pair0,2 eliminate row0 pivot1 entry1 product = 1

directsum pair0,2 eliminate row0 pivot1 entry1 difference := sub(1, 1)

directsum pair0,2 eliminate row0 pivot1 entry1 difference = 0

directsum pair0,2:target row=0;old=[1, 1];factor=1;pivot row=[0, 1]

new full row=[1, 0]

directsum pair0,2:complete exact reduced matrix=[[1, 0], [0, 1]];pivot columns=[0, 1]

rank=2;pivot columns independent and all original column relations preserved;zero rows retained

directsum pair0,2 basis counts := sum(1, 1)

directsum pair0,2 basis counts = 2

directsum pair0,2 intersection dimension := sub(2, 2)

directsum pair0,2 intersection dimension = 0

pair original subspaces=[0, 2];individual independent basis counts=[1, 1];combined rank=2

dim intersection=0;pair intersection zero=是

directsum pair1,2:complete ordered matrix=[[0, 1], [1, 13]];rows=2;columns=2

use exact invertible ROW operations only;original COLUMN roles/order stay fixed

directsum pair1,2:matrix=[[0, 1], [1, 13]];selected row=1

swap rows0,1;matrix=[[1, 13], [0, 1]]

directsum pair1,2 pivot0 column0 normalized entry0 := div(1, 1)

directsum pair1,2 pivot0 column0 normalized entry0 = 1

directsum pair1,2 pivot0 column0 normalized entry1 := div(13, 1)

directsum pair1,2 pivot0 column0 normalized entry1 = 13

directsum pair1,2:pivot row=0;column=0;nonzero pivot=1;old row=[1, 13]

normalized full row=[1, 13]

directsum pair1,2 pivot1 column1 normalized entry0 := div(0, 1)

directsum pair1,2 pivot1 column1 normalized entry0 = 0

directsum pair1,2 pivot1 column1 normalized entry1 := div(1, 1)

directsum pair1,2 pivot1 column1 normalized entry1 = 1

directsum pair1,2:pivot row=1;column=1;nonzero pivot=1;old row=[0, 1]

normalized full row=[0, 1]

directsum pair1,2 eliminate row0 pivot1 entry0 product := mul(13, 0)

directsum pair1,2 eliminate row0 pivot1 entry0 product = 0

directsum pair1,2 eliminate row0 pivot1 entry0 difference := sub(1, 0)

directsum pair1,2 eliminate row0 pivot1 entry0 difference = 1

directsum pair1,2 eliminate row0 pivot1 entry1 product := mul(13, 1)

directsum pair1,2 eliminate row0 pivot1 entry1 product = 13

directsum pair1,2 eliminate row0 pivot1 entry1 difference := sub(13, 13)

directsum pair1,2 eliminate row0 pivot1 entry1 difference = 0

directsum pair1,2:target row=0;old=[1, 13];factor=13;pivot row=[0, 1]

new full row=[1, 0]

directsum pair1,2:complete exact reduced matrix=[[1, 0], [0, 1]];pivot columns=[0, 1]

rank=2;pivot columns independent and all original column relations preserved;zero rows retained

directsum pair1,2 basis counts := sum(1, 1)

directsum pair1,2 basis counts = 2

directsum pair1,2 intersection dimension := sub(2, 2)

directsum pair1,2 intersection dimension = 0

pair original subspaces=[1, 2];individual independent basis counts=[1, 1];combined rank=2

dim intersection=0;pair intersection zero=是

directsum ALL subspace columns:complete ordered matrix=[[1, 0, 1], [0, 1, 13]];rows=2;columns=3

use exact invertible ROW operations only;original COLUMN roles/order stay fixed

directsum ALL subspace columns pivot0 column0 normalized entry0 := div(1, 1)

directsum ALL subspace columns pivot0 column0 normalized entry0 = 1

directsum ALL subspace columns pivot0 column0 normalized entry1 := div(0, 1)

directsum ALL subspace columns pivot0 column0 normalized entry1 = 0

directsum ALL subspace columns pivot0 column0 normalized entry2 := div(1, 1)

directsum ALL subspace columns pivot0 column0 normalized entry2 = 1

directsum ALL subspace columns:pivot row=0;column=0;nonzero pivot=1;old row=[1, 0, 1]

normalized full row=[1, 0, 1]

directsum ALL subspace columns pivot1 column1 normalized entry0 := div(0, 1)

directsum ALL subspace columns pivot1 column1 normalized entry0 = 0

directsum ALL subspace columns pivot1 column1 normalized entry1 := div(1, 1)

directsum ALL subspace columns pivot1 column1 normalized entry1 = 1

directsum ALL subspace columns pivot1 column1 normalized entry2 := div(13, 1)

directsum ALL subspace columns pivot1 column1 normalized entry2 = 13

directsum ALL subspace columns:pivot row=1;column=1;nonzero pivot=1;old row=[0, 1, 13]

normalized full row=[0, 1, 13]

directsum ALL subspace columns:complete exact reduced matrix=[[1, 0, 1], [0, 1, 13]];pivot columns=[0, 1]

rank=2;pivot columns independent and all original column relations preserved;zero rows retained

directsum ALL subspaces free2 pivot0 relation coefficient := mul(-1, 1)

directsum ALL subspaces free2 pivot0 relation coefficient = -1

directsum ALL subspaces free2 pivot1 relation coefficient := mul(-1, 13)

directsum ALL subspaces free2 pivot1 relation coefficient = -13

directsum ALL subspaces relation free2 original coordinate0 coordinate0 product := mul(1, -1)

directsum ALL subspaces relation free2 original coordinate0 coordinate0 product = -1

directsum ALL subspaces relation free2 original coordinate0 coordinate1 product := mul(0, -13)

directsum ALL subspaces relation free2 original coordinate0 coordinate1 product = 0

directsum ALL subspaces relation free2 original coordinate0 coordinate2 product := mul(1, 1)

directsum ALL subspaces relation free2 original coordinate0 coordinate2 product = 1

directsum ALL subspaces relation free2 original coordinate0 complete sum := sum(-1, 0, 1)

directsum ALL subspaces relation free2 original coordinate0 complete sum = 0

directsum ALL subspaces relation free2 original coordinate1 coordinate0 product := mul(0, -1)

directsum ALL subspaces relation free2 original coordinate1 coordinate0 product = 0

directsum ALL subspaces relation free2 original coordinate1 coordinate1 product := mul(1, -13)

directsum ALL subspaces relation free2 original coordinate1 coordinate1 product = -13

directsum ALL subspaces relation free2 original coordinate1 coordinate2 product := mul(13, 1)

directsum ALL subspaces relation free2 original coordinate1 coordinate2 product = 13

directsum ALL subspaces relation free2 original coordinate1 complete sum := sum(0, -13, 13)

directsum ALL subspaces relation free2 original coordinate1 complete sum = 0

directsum ALL subspaces:free column=2;original ordered columns=[[1, 0], [0, 1], [1, 13]];exact relation coefficients=[-1, -13, 1]

nonzero coefficient at free position;full original combination=[0, 0]

directsum ALL subspaces:pivot columns=[0, 1];ALL free columns=[2];full relation family=[[-1, -13, 1]]

EVERY zero combination is uniquely determined by its free coordinates;these free-coordinate unit relations form the full kernel;dimension=1

directsum total ordered basis count := sum(1, 1, 1)

directsum total ordered basis count = 3

ALL pair intersection dimensions=[[0, 1, 0], [0, 2, 0], [1, 2, 0]];pairwise zero=是;GLOBAL original ordered basis count=3;global rank=2;full relations=[[-1, -13, 1]]

global directness=否 from injectivity of the FULL sum map;ambient coverage=是 separately;pairwise_zero_not_sufficient_for_three_or_more

逐步反向核對

• S088 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S087 · premise_and_result_audit — 以精確有理數重算 directsum total ordered basis count，不以小數近似替代。
• S086 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S085 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S084 · premise_and_result_audit — 以精確有理數重算 directsum ALL subspaces relation free2 original coordinate1 complete sum，不以小數近似替代。
• S083 · premise_and_result_audit — 以精確有理數重算 directsum ALL subspaces relation free2 original coordinate1 coordinate2 product，不以小數近似替代。
• S082 · premise_and_result_audit — 以精確有理數重算 directsum ALL subspaces relation free2 original coordinate1 coordinate1 product，不以小數近似替代。
• S081 · premise_and_result_audit — 以精確有理數重算 directsum ALL subspaces relation free2 original coordinate1 coordinate0 product，不以小數近似替代。
• S080 · premise_and_result_audit — 以精確有理數重算 directsum ALL subspaces relation free2 original coordinate0 complete sum，不以小數近似替代。
• S079 · premise_and_result_audit — 以精確有理數重算 directsum ALL subspaces relation free2 original coordinate0 coordinate2 product，不以小數近似替代。
• S078 · premise_and_result_audit — 以精確有理數重算 directsum ALL subspaces relation free2 original coordinate0 coordinate1 product，不以小數近似替代。
• S077 · premise_and_result_audit — 以精確有理數重算 directsum ALL subspaces relation free2 original coordinate0 coordinate0 product，不以小數近似替代。
• S076 · premise_and_result_audit — 以精確有理數重算 directsum ALL subspaces free2 pivot1 relation coefficient，不以小數近似替代。
• S075 · premise_and_result_audit — 以精確有理數重算 directsum ALL subspaces free2 pivot0 relation coefficient，不以小數近似替代。
• S074 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S073 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S072 · premise_and_result_audit — 以精確有理數重算 directsum ALL subspace columns pivot1 column1 normalized entry2，不以小數近似替代。
• S071 · premise_and_result_audit — 以精確有理數重算 directsum ALL subspace columns pivot1 column1 normalized entry1，不以小數近似替代。
• S070 · premise_and_result_audit — 以精確有理數重算 directsum ALL subspace columns pivot1 column1 normalized entry0，不以小數近似替代。
• S069 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S068 · premise_and_result_audit — 以精確有理數重算 directsum ALL subspace columns pivot0 column0 normalized entry2，不以小數近似替代。
• S067 · premise_and_result_audit — 以精確有理數重算 directsum ALL subspace columns pivot0 column0 normalized entry1，不以小數近似替代。
• S066 · premise_and_result_audit — 以精確有理數重算 directsum ALL subspace columns pivot0 column0 normalized entry0，不以小數近似替代。
• S065 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S064 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S063 · premise_and_result_audit — 以精確有理數重算 directsum pair1,2 intersection dimension，不以小數近似替代。
• S062 · premise_and_result_audit — 以精確有理數重算 directsum pair1,2 basis counts，不以小數近似替代。
• S061 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S060 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S059 · premise_and_result_audit — 以精確有理數重算 directsum pair1,2 eliminate row0 pivot1 entry1 difference，不以小數近似替代。
• S058 · premise_and_result_audit — 以精確有理數重算 directsum pair1,2 eliminate row0 pivot1 entry1 product，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 directsum pair1,2 eliminate row0 pivot1 entry0 difference，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 directsum pair1,2 eliminate row0 pivot1 entry0 product，不以小數近似替代。
• S055 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S054 · premise_and_result_audit — 以精確有理數重算 directsum pair1,2 pivot1 column1 normalized entry1，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 directsum pair1,2 pivot1 column1 normalized entry0，不以小數近似替代。
• S052 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S051 · premise_and_result_audit — 以精確有理數重算 directsum pair1,2 pivot0 column0 normalized entry1，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 directsum pair1,2 pivot0 column0 normalized entry0，不以小數近似替代。
• S049 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S048 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S047 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S046 · premise_and_result_audit — 以精確有理數重算 directsum pair0,2 intersection dimension，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 directsum pair0,2 basis counts，不以小數近似替代。
• S044 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S043 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S042 · premise_and_result_audit — 以精確有理數重算 directsum pair0,2 eliminate row0 pivot1 entry1 difference，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 directsum pair0,2 eliminate row0 pivot1 entry1 product，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 directsum pair0,2 eliminate row0 pivot1 entry0 difference，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 directsum pair0,2 eliminate row0 pivot1 entry0 product，不以小數近似替代。
• S038 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S037 · premise_and_result_audit — 以精確有理數重算 directsum pair0,2 pivot1 column1 normalized entry1，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 directsum pair0,2 pivot1 column1 normalized entry0，不以小數近似替代。
• S035 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S034 · premise_and_result_audit — 以精確有理數重算 directsum pair0,2 pivot0 column0 normalized entry1，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 directsum pair0,2 pivot0 column0 normalized entry0，不以小數近似替代。
• S032 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S031 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S030 · premise_and_result_audit — 以精確有理數重算 directsum pair0,1 intersection dimension，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 directsum pair0,1 basis counts，不以小數近似替代。
• S028 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S027 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S026 · premise_and_result_audit — 以精確有理數重算 directsum pair0,1 pivot1 column1 normalized entry1，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 directsum pair0,1 pivot1 column1 normalized entry0，不以小數近似替代。
• S024 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S023 · premise_and_result_audit — 以精確有理數重算 directsum pair0,1 pivot0 column0 normalized entry1，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 directsum pair0,1 pivot0 column0 normalized entry0，不以小數近似替代。
• S021 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S020 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S019 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S018 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S017 · premise_and_result_audit — 以精確有理數重算 directsum subspace2 eliminate row1 pivot0 entry0 difference，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 directsum subspace2 eliminate row1 pivot0 entry0 product，不以小數近似替代。
• S015 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S014 · premise_and_result_audit — 以精確有理數重算 directsum subspace2 pivot0 column0 normalized entry0，不以小數近似替代。
• S013 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S012 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S011 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S010 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S009 · premise_and_result_audit — 以精確有理數重算 directsum subspace1 pivot0 column0 normalized entry0，不以小數近似替代。
• S008 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S007 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S006 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S005 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S003 · premise_and_result_audit — 以精確有理數重算 directsum subspace0 pivot0 column0 normalized entry0，不以小數近似替代。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

(是, 3, 2, 否, 是, pairwise_zero_not_sufficient_for_three_or_more)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。